The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction

ABSTRACT The directed Oberwolfach problem asks whether the complete symmetric digraph , assuming , admits a decomposition into spanning subdigraphs, each a disjoint union of directed cycles of lengths . We hereby describe a method for constructing a solution to given a solution to , for some , if certain conditions on are satisfied. This approach enables us to extend a solution for into a solution for , as well as into a solution for , where denotes copies of 2, provided is sufficiently large. In particular, our recursive construction allows us to effectively address the two‐table directed Oberwolfach problem. We show that has a solution for all , with a definite exception of and a possible exception in the case that , is even, and . It has been shown previously that has a solution if is odd, and that has a solution if and only if . In addition to solving many other cases of , we show that when , has a solution if and only if .